Elasticity statistics

Kaliske, M., Heinrich, G., 1999. An extended tube-model for rubber elasticity Statistical-mechanical theory and finite element implementation. Rubber Chem. Technol. 72 (4), 602-632. Khokhlov, A.R., 1992. In Dusek, K. (Ed.), Responsive Gels Volume Transitions I. Springer, Verlag Berlin, p. 125. 

Kaliske, M. and Heinrich, G. (1999) An extended tube-model for rubber elasticity statistical-mechanical theory and finite element implementation. Rubber Chem. 

Carpick et al [M] used AFM, with a Pt-coated tip on a mica substrate in ultraliigh vacuum, to show that if the defonnation of the substrate and the tip-substrate adhesion are taken into account (the so-called JKR model [175] of elastic adliesive contact), then the frictional force is indeed proportional to the contact area between tip and sample. Flowever, under these smgle-asperity conditions, Amontons law does not hold, since the statistical effect of more asperities coming into play no longer occurs, and the contact area is not simply proportional to the applied load. 

Another simplified model is the freely jointed or random flight chain model. It assumes all bond and conformation angles can have any value with no energy penalty, and gives a simplified statistical description of elasticity and average end-to-end distance. 

We shall rely heavily on models again in this chapter this time they are of two different types. We shall consider elasticity in terms of a molecular model in which the chains are described by random flight statistics. The phenomena of... 

It is not particularly difficult to introduce thermodynamic concepts into a discussion of elasticity. We shall not explore all of the implications of this development, but shall proceed only to the point of establishing the connection between elasticity and entropy. Then we shall go from phenomenological thermodynamics to statistical thermodynamics in pursuit of a molecular model to describe the elastic response of cross-linked networks. 

A great many liquids have entropies of vaporization at the normal boiling point in the vicinity of this value (see benzene above), a generalization known as Trouton s rule. Our interest is clearly not in evaporation, but in the elongation of elastomers. In the next section we shall apply Eq. (3.21) to the stretching process for a statistical—and therefore molecular—picture of elasticity. 

Thermal Properties at Low Temperatures For sohds, the Debye model developed with the aid of statistical mechanics and quantum theoiy gives a satisfactoiy representation of the specific heat with temperature. Procedures for calculating values of d, ihe Debye characteristic temperature, using either elastic constants, the compressibility, the melting point, or the temperature dependence of the expansion coefficient are outlined by Barron (Cryogenic Systems, 2d ed., Oxford University Press, 1985, pp 24-29). 

J. H. Weiner. Statistical Mechanics of Elasticity. New York Wiley, 1983. 

To understand the global mechanical and statistical properties of polymeric systems as well as studying the conformational relaxation of melts and amorphous systems, it is important to go beyond the atomistic level. One of the central questions of the physics of polymer melts and networks throughout the last 20 years or so dealt with the role of chain topology for melt dynamics and the elastic modulus of polymer networks. The fact that the different polymer strands cannot cut through each other in the... 

To be more precise, let us assume, as Boltzman first did in 1872 [boltz72], that we have N perfectly elastic billiard balls, or hard-spheres, inside a volume V, and that a complete statistical description of our system (be it a gas or fluid) at, or near, its equilibrium state is contained in the one-particle phase-space distribution function f x,v,t) ... 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The bracket (1 — 2/f) was introduced into the theory of rubber elasticity by Graessley [23], following an idea of Duiser and Staverman [28]. Graessley discussed the statistical mechanics of random coil networks, which he had divided into an ensemble of micronetworks. 

It is also possible to estimate the cross-link density from the stress-strain data, using the statistical theory of rubber-like elasticity [47,58]. For a swollen rubber the relationship is... 

The entropic elasticity of a single polymer chain is treated similarly to the statistical mechanical property of gas molecules. Now, let us compare gas molecules to children moving around freely... 

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

This apparent time dependent cell disruption is caused because of the statistically random distribution of the orientation of the cells within a flow field and the random changes in that distribution as a function of time, the latter is caused as the cells spin in the flow field in response to the forces that act on them. In the present discussion this is referred to as apparent time dependency in order to distinguish it from true time-dependent disruption arising from anelastic behaviour of the cell walls. Anelastic behaviour, or time-dependent elasticity, is thought to arise from a restructuring of the fabric of the cell wall material at a molecular level. Anelasticity is stress induced and requires energy which is dissipated as heat, and if it is excessive it can weaken the structure and cause its breakage. 

The statistical distribution of r values for long polymer chains and the influence of chain structure and hindrance to rotation about chain bonds on its root-mean-square value will be the topics of primary concern in the present chapter. We thus enter upon the second major application of statistical methods to polymer problems, the first of these having been discussed in the two chapters preceding. Quite apart from whatever intrinsic interest may be attached to the polymer chain configuration problem, its analysis is essential for the interpretation of rubberlike elasticity and of dilute solution properties, both hydrodynamic and thermodynamic, of polymers. These problems will be dealt with in following chapters. The content of the present... 

The formal thermodynamic analogy existing between an ideal rubber and an ideal gas carries over to the statistical derivation of the force of retraction of stretched rubber, which we undertake in this section. This derivation parallels so closely the statistical-thermodynamic deduction of the pressure of a perfect gas that it seems worth while to set forth the latter briefly here for the purpose of illustrating clearly the subsequent derivation of the basic relations of rubber elasticity theory. 

This is Mooney s equation for the stored elastic energy per unit volume. The constant Ci corresponds to the kTvel V of the statistical theory i.e., the first term in Eq. (49) is of the same form as the theoretical elastic free energy per unit volume AF =—TAiS/F where AaS is given by Eq. (41) with axayaz l. The second term in Eq. (49) contains the parameter whose significance from the point of view of the structure of the elastic body remains unknown at present. For simple extension, ax = a, ay — az—X/a, and the retractive force r per unit initial cross section, given by dW/da, is... 

---